The Journal of Geometric Mechanics (JGM)

Variational principles for spin systems and the Kirchhoff rod

Pages: 417 - 444, Volume 1, Issue 4, December 2009      doi:10.3934/jgm.2009.1.417

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François Gay-Balma - Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 911125, United States (email)
Darryl D. Holm - Department of Mathematics and Institute for Mathematical Sciences, Imperial College, London, SW7 2AZ, United Kingdom (email)
Tudor S. Ratiu - Section de Mathématiques and Bernoulli Center, Ecole Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland (email)

Abstract: We obtain the affine Euler-Poincaré equations by standard Lagrangian reduction and deduce the associated Clebsch-constrained variational principle. These results are illustrated in deriving the equations of motion for continuum spin systems and Kirchhoff's rod, where they provide a unified geometric interpretation.

Keywords:  variational principles, symmetry, Lagrangian reducton, cocycle, affine Euler-Poincaré equations, spin system, Kirchhoff rod.
Mathematics Subject Classification:  Primary: 70H33, 70G65, 70G75; Secondary: 74K10, 82D30.

Received: April 2009;      Revised: July 2009;      Available Online: January 2010.